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If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4

Thanks! 

Dimitristhym  Oct 13, 2018
 #1
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"If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4"

 

.

Alan  Oct 13, 2018
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You assumed that A is an invertible matrix

Guest Oct 13, 2018
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Ohh Thank you very much Alan!Have a nice day!

Dimitristhym  Oct 13, 2018
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If it's not we can't prove Α^-1 = -Α^4 

Dimitristhym  Oct 13, 2018
 #5
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You know the question better than me, are you supposed to assume that A is invertible? It IS possible to prove that A is invertible, so I don't think you're supposed to assume that.

Guest Oct 13, 2018
 #6
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First you must prove that,A is invertible because if it don't we can't prove Α^-1 = -Α^4

Dimitristhym  Oct 13, 2018
 #7
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I know that, but I'm not going to give you the answer, I want you to try first

Guest Oct 13, 2018
 #8
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I think i know how to prove it im in 2nd year in Greek university in Math science but this is a lesson for 1st year 

and I'm doing it again!

Dimitristhym  Oct 13, 2018
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Its -A^4 + A^3 - A^2 + A = I <=> 

<=> A( -A^3 + A^4 - A + I) =I 

so ( -A^3 + A^4 - A + I)=A^-1

Dimitristhym  Oct 13, 2018
 #10
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Yes I think you got it, But just to be sure:

 

You got the equation A*(-A3+A2-A1+I)=I. We can replace -A3+A2-A1+I with "B". So we got A*B=I. I is an invertible matrix, and if a multiplication of two matrices results in an invertible matrix this means that both matrices are also invertible, meaning that A is invertible.

Guest Oct 13, 2018
 #11
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Yes exactly from definition of  invertible matrix!

Thanks guys for your tips! 

Dimitristhym  Oct 13, 2018

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